Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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Concept of Limits - Algebra of Limits

notes

The limiting process respects addition, subtraction, multiplication and division as long as the limits and functions under consideration are well defined. In fact, below we formalise these as a theorem without proof.

theorem

Theorem: 
 Let f  and g be two functions such that both `lim_(x -> a)` f(x) and `lim_(x -> a)` g(x) exist.
Then

(i) Limit of sum of two functions is sum of the limits of the functions, i.e.,
`lim_(x -> a) [f(x) + g(x)]` = `lim_(x -> a) f(x) + lim _(x -> a) g(x)`.

(ii) Limit of difference of two functions is difference of the limits of the functions, i.e.,
`lim_(x -> a) [f(x) -g(x)] = lim_(x -> a) f(x) -lim _(x -> a) g(x)`.

(iii) Limit of product of two functions is product of the limits of the functions, i.e.,
`lim_(x -> a) [f(x)  . g(x)] = lim_(x -> a) f(x)  .  lim _(x -> a) g(x)`.

(iv) Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero), i.e., 
`lim_(x -> a) f(x)/g(x)  =(lim_(x->a) f(x))/(lim_(x-> a )  g(x))`

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